HARMONIC ANALYSIS! AND PREDICTION OF TIDES 149 
in which 
H,=|H2+ H}?—2H,H, cos (G,—G,))? (470) 
_, H, sin G.—H, sin G, 
H, cos G,—H, cos G, 
The proper quadrant for G,z is determined by the signs of the numer- 
ator and denominator of the above fraction, these being the same, 
respectively, as for the sine and cosine of the angle. Formulas (470) 
and (471) may be solved graphically (fig. 34) by drawing from any 
point C a line CD to represent in length and direction H, and G,, 
respectively; from the point D a line DE to represent in length and 
direction H, and (G,+180°), respectively. The connecting line from 
Ga=tan (471) 
90° 
180 
FIGURE 34. 
C to FE’ will represent by its length the amplitude H, and by its diree- 
tion the epoch G). 
440. Formulas (470) and (471) may be modified to adapt them for 
use with tables 41 and 42. 
From (470) we may obtain 
Ala/Ha=([1 + (A/Ha)’+ 2(Hp/H_) cos (Gy—Ga+180°)]} (472) 
A1,/H,=[1 + (H./He)?+2(Ha/Hr) cos (Ga—Gr+180°)}! (478) 
and from (471) we have 
(H,/H,) sin (G,—G,+£180°) 
or 
